Problem: How many positive integers $n$ satisfy $200 < n^2 < 900$?
Since $f(n)=n^2$ is a monotonically increasing function (on the set of positive integers), we can find the least and greatest integer solutions and count the integers between them.  Since $14^2=196$ and $15^2=225$, $n=15$ is the smallest solution.  Since $30^2=900$, $n=29$ is the largest solution.  There are $29-15+1=\boxed{15}$ integers between 15 and 29 inclusive.